Families#

A Family is an associative container which models a family \((f_i)_{i \in I}\). Then, f[i] returns the element of the family indexed by i. Whenever available, set and combinatorial class operations (counting, iteration, listing) on the family are induced from those of the index set. Families should be created through the Family() function.

AUTHORS:

  • Nicolas Thiery (2008-02): initial release

  • Florent Hivert (2008-04): various fixes, cleanups and improvements.

class sage.sets.family.AbstractFamily[source]#

Bases: Parent

The abstract class for family

Any family belongs to a class which inherits from AbstractFamily.

hidden_keys()[source]#

Returns the hidden keys of the family, if any.

EXAMPLES:

sage: f = Family({3: 'a', 4: 'b', 7: 'd'})
sage: f.hidden_keys()
[]

>>> from sage.all import *
>>> f = Family({Integer(3): 'a', Integer(4): 'b', Integer(7): 'd'})
>>> f.hidden_keys()
[]
inverse_family()[source]#

Returns the inverse family, with keys and values exchanged. This presumes that there are no duplicate values in self.

This default implementation is not lazy and therefore will only work with not too big finite families. It is also cached for the same reason:

sage: Family({3: 'a', 4: 'b', 7: 'd'}).inverse_family()
Finite family {'a': 3, 'b': 4, 'd': 7}

sage: Family((3,4,7)).inverse_family()
Finite family {3: 0, 4: 1, 7: 2}

>>> from sage.all import *
>>> Family({Integer(3): 'a', Integer(4): 'b', Integer(7): 'd'}).inverse_family()
Finite family {'a': 3, 'b': 4, 'd': 7}

>>> Family((Integer(3),Integer(4),Integer(7))).inverse_family()
Finite family {3: 0, 4: 1, 7: 2}
items()[source]#

Return an iterator for key-value pairs.

A key can only appear once, but if the function is not injective, values may appear multiple times.

EXAMPLES:

sage: f = Family([-2, -1, 0, 1, 2], abs)
sage: list(f.items())
[(-2, 2), (-1, 1), (0, 0), (1, 1), (2, 2)]

>>> from sage.all import *
>>> f = Family([-Integer(2), -Integer(1), Integer(0), Integer(1), Integer(2)], abs)
>>> list(f.items())
[(-2, 2), (-1, 1), (0, 0), (1, 1), (2, 2)]
keys()[source]#

Return the keys of the family.

EXAMPLES:

sage: f = Family({3: 'a', 4: 'b', 7: 'd'})
sage: sorted(f.keys())
[3, 4, 7]

>>> from sage.all import *
>>> f = Family({Integer(3): 'a', Integer(4): 'b', Integer(7): 'd'})
>>> sorted(f.keys())
[3, 4, 7]
map(f, name=None)[source]#

Return the family \(( f(\mathtt{self}[i]) )_{i \in I}\), where \(I\) is the index set of self.

Todo

good name?

EXAMPLES:

sage: f = Family({3: 'a', 4: 'b', 7: 'd'})
sage: g = f.map(lambda x: x+'1')
sage: list(g)
['a1', 'b1', 'd1']

>>> from sage.all import *
>>> f = Family({Integer(3): 'a', Integer(4): 'b', Integer(7): 'd'})
>>> g = f.map(lambda x: x+'1')
>>> list(g)
['a1', 'b1', 'd1']
values()[source]#

Return the elements (values) of this family.

EXAMPLES:

sage: f = Family(["c", "a", "b"], lambda x: x + x)
sage: sorted(f.values())
['aa', 'bb', 'cc']

>>> from sage.all import *
>>> f = Family(["c", "a", "b"], lambda x: x + x)
>>> sorted(f.values())
['aa', 'bb', 'cc']
zip(f, other, name=None)[source]#

Given two families with same index set \(I\) (and same hidden keys if relevant), returns the family \(( f(self[i], other[i]) )_{i \in I}\)

Todo

generalize to any number of families and merge with map?

EXAMPLES:

sage: f = Family({3: 'a', 4: 'b', 7: 'd'})
sage: g = Family({3: '1', 4: '2', 7: '3'})
sage: h = f.zip(lambda x,y: x+y, g)
sage: list(h)
['a1', 'b2', 'd3']

>>> from sage.all import *
>>> f = Family({Integer(3): 'a', Integer(4): 'b', Integer(7): 'd'})
>>> g = Family({Integer(3): '1', Integer(4): '2', Integer(7): '3'})
>>> h = f.zip(lambda x,y: x+y, g)
>>> list(h)
['a1', 'b2', 'd3']
class sage.sets.family.EnumeratedFamily(enumset)[source]#

Bases: LazyFamily

EnumeratedFamily turns an enumerated set c into a family indexed by the set \(\{0,\dots, |c|-1\}\) (or NN if \(|c|\) is countably infinite).

Instances should be created via the Family() factory. See its documentation for examples and tests.

cardinality()[source]#

Return the number of elements in self.

EXAMPLES:

sage: from sage.sets.family import EnumeratedFamily
sage: f = EnumeratedFamily(Permutations(3))
sage: f.cardinality()
6

sage: f = Family(NonNegativeIntegers())
sage: f.cardinality()
+Infinity

>>> from sage.all import *
>>> from sage.sets.family import EnumeratedFamily
>>> f = EnumeratedFamily(Permutations(Integer(3)))
>>> f.cardinality()
6

>>> f = Family(NonNegativeIntegers())
>>> f.cardinality()
+Infinity
sage.sets.family.Family(indices, function=None, hidden_keys=[], hidden_function=None, lazy=False, name=None)[source]#

A Family is an associative container which models a family \((f_i)_{i \in I}\). Then, f[i] returns the element of the family indexed by \(i\). Whenever available, set and combinatorial class operations (counting, iteration, listing) on the family are induced from those of the index set.

There are several available implementations (classes) for different usages; Family serves as a factory, and will create instances of the appropriate classes depending on its arguments.

INPUT:

  • indices – the indices for the family

  • function – (optional) the function \(f\) applied to all visible indices; the default is the identity function

  • hidden_keys – (optional) a list of hidden indices that can be accessed through my_family[i]

  • hidden_function – (optional) a function for the hidden indices

  • lazy – boolean (default: False); whether the family is lazily created or not; if the indices are infinite, then this is automatically made True

  • name – (optional) the name of the function; only used when the family is lazily created via a function

EXAMPLES:

In its simplest form, a list \(l = [l_0, l_1, \ldots, l_{\ell}]\) or a tuple by itself is considered as the family \((l_i)_{i \in I}\) where \(I\) is the set \(\{0, \ldots, \ell\}\) where \(\ell\) is len(l) - 1. So Family(l) returns the corresponding family:

sage: f = Family([1,2,3])
sage: f
Family (1, 2, 3)
sage: f = Family((1,2,3))
sage: f
Family (1, 2, 3)

>>> from sage.all import *
>>> f = Family([Integer(1),Integer(2),Integer(3)])
>>> f
Family (1, 2, 3)
>>> f = Family((Integer(1),Integer(2),Integer(3)))
>>> f
Family (1, 2, 3)

Instead of a list you can as well pass any iterable object:

sage: f = Family(2*i+1 for i in [1,2,3])
sage: f
Family (3, 5, 7)

>>> from sage.all import *
>>> f = Family(Integer(2)*i+Integer(1) for i in [Integer(1),Integer(2),Integer(3)])
>>> f
Family (3, 5, 7)

A family can also be constructed from a dictionary t. The resulting family is very close to t, except that the elements of the family are the values of t. Here, we define the family \((f_i)_{i \in \{3,4,7\}}\) with \(f_3 = a\), \(f_4 = b\), and \(f_7 = d\):

sage: f = Family({3: 'a', 4: 'b', 7: 'd'})
sage: f
Finite family {3: 'a', 4: 'b', 7: 'd'}
sage: f[7]
'd'
sage: len(f)
3
sage: list(f)
['a', 'b', 'd']
sage: [ x for x in f ]
['a', 'b', 'd']
sage: f.keys()
[3, 4, 7]
sage: 'b' in f
True
sage: 'e' in f
False

>>> from sage.all import *
>>> f = Family({Integer(3): 'a', Integer(4): 'b', Integer(7): 'd'})
>>> f
Finite family {3: 'a', 4: 'b', 7: 'd'}
>>> f[Integer(7)]
'd'
>>> len(f)
3
>>> list(f)
['a', 'b', 'd']
>>> [ x for x in f ]
['a', 'b', 'd']
>>> f.keys()
[3, 4, 7]
>>> 'b' in f
True
>>> 'e' in f
False

A family can also be constructed by its index set \(I\) and a function \(f\), as in \((f(i))_{i \in I}\):

sage: f = Family([3,4,7], lambda i: 2*i)
sage: f
Finite family {3: 6, 4: 8, 7: 14}
sage: f.keys()
[3, 4, 7]
sage: f[7]
14
sage: list(f)
[6, 8, 14]
sage: [x for x in f]
[6, 8, 14]
sage: len(f)
3

>>> from sage.all import *
>>> f = Family([Integer(3),Integer(4),Integer(7)], lambda i: Integer(2)*i)
>>> f
Finite family {3: 6, 4: 8, 7: 14}
>>> f.keys()
[3, 4, 7]
>>> f[Integer(7)]
14
>>> list(f)
[6, 8, 14]
>>> [x for x in f]
[6, 8, 14]
>>> len(f)
3

By default, all images are computed right away, and stored in an internal dictionary:

sage: f = Family((3,4,7), lambda i: 2*i)
sage: f
Finite family {3: 6, 4: 8, 7: 14}

>>> from sage.all import *
>>> f = Family((Integer(3),Integer(4),Integer(7)), lambda i: Integer(2)*i)
>>> f
Finite family {3: 6, 4: 8, 7: 14}

Note that this requires all the elements of the list to be hashable. One can ask instead for the images \(f(i)\) to be computed lazily, when needed:

sage: f = Family([3,4,7], lambda i: 2*i, lazy=True)
sage: f
Lazy family (<lambda>(i))_{i in [3, 4, 7]}
sage: f[7]
14
sage: list(f)
[6, 8, 14]
sage: [x for x in f]
[6, 8, 14]

>>> from sage.all import *
>>> f = Family([Integer(3),Integer(4),Integer(7)], lambda i: Integer(2)*i, lazy=True)
>>> f
Lazy family (<lambda>(i))_{i in [3, 4, 7]}
>>> f[Integer(7)]
14
>>> list(f)
[6, 8, 14]
>>> [x for x in f]
[6, 8, 14]

This allows in particular for modeling infinite families:

sage: f = Family(ZZ, lambda i: 2*i, lazy=True)
sage: f
Lazy family (<lambda>(i))_{i in Integer Ring}
sage: f.keys()
Integer Ring
sage: f[1]
2
sage: f[-5]
-10
sage: i = iter(f)
sage: next(i), next(i), next(i), next(i), next(i)
(0, 2, -2, 4, -4)

>>> from sage.all import *
>>> f = Family(ZZ, lambda i: Integer(2)*i, lazy=True)
>>> f
Lazy family (<lambda>(i))_{i in Integer Ring}
>>> f.keys()
Integer Ring
>>> f[Integer(1)]
2
>>> f[-Integer(5)]
-10
>>> i = iter(f)
>>> next(i), next(i), next(i), next(i), next(i)
(0, 2, -2, 4, -4)

Note that the lazy keyword parameter is only needed to force laziness. Usually it is automatically set to a correct default value (ie: False for finite data structures and True for enumerated sets:

sage: f == Family(ZZ, lambda i: 2*i)
True

>>> from sage.all import *
>>> f == Family(ZZ, lambda i: Integer(2)*i)
True

Beware that for those kind of families len(f) is not supposed to work. As a replacement, use the .cardinality() method:

sage: f = Family(Permutations(3), attrcall("to_lehmer_code"))
sage: list(f)
[[0, 0, 0], [0, 1, 0], [1, 0, 0], [1, 1, 0], [2, 0, 0], [2, 1, 0]]
sage: f.cardinality()
6

>>> from sage.all import *
>>> f = Family(Permutations(Integer(3)), attrcall("to_lehmer_code"))
>>> list(f)
[[0, 0, 0], [0, 1, 0], [1, 0, 0], [1, 1, 0], [2, 0, 0], [2, 1, 0]]
>>> f.cardinality()
6

Caveat: Only certain families with lazy behavior can be pickled. In particular, only functions that work with Sage’s pickle_function and unpickle_function (in sage.misc.fpickle) will correctly unpickle. The following two work:

sage: f = Family(Permutations(3), lambda p: p.to_lehmer_code()); f
Lazy family (<lambda>(i))_{i in Standard permutations of 3}
sage: f == loads(dumps(f))
True

sage: f = Family(Permutations(3), attrcall("to_lehmer_code")); f
Lazy family (i.to_lehmer_code())_{i in Standard permutations of 3}
sage: f == loads(dumps(f))
True

>>> from sage.all import *
>>> f = Family(Permutations(Integer(3)), lambda p: p.to_lehmer_code()); f
Lazy family (<lambda>(i))_{i in Standard permutations of 3}
>>> f == loads(dumps(f))
True

>>> f = Family(Permutations(Integer(3)), attrcall("to_lehmer_code")); f
Lazy family (i.to_lehmer_code())_{i in Standard permutations of 3}
>>> f == loads(dumps(f))
True

But this one does not:

sage: def plus_n(n): return lambda x: x+n
sage: f = Family([1,2,3], plus_n(3), lazy=True); f
Lazy family (<lambda>(i))_{i in [1, 2, 3]}
sage: f == loads(dumps(f))
Traceback (most recent call last):
...
ValueError: Cannot pickle code objects from closures

>>> from sage.all import *
>>> def plus_n(n): return lambda x: x+n
>>> f = Family([Integer(1),Integer(2),Integer(3)], plus_n(Integer(3)), lazy=True); f
Lazy family (<lambda>(i))_{i in [1, 2, 3]}
>>> f == loads(dumps(f))
Traceback (most recent call last):
...
ValueError: Cannot pickle code objects from closures

Finally, it can occasionally be useful to add some hidden elements in a family, which are accessible as f[i], but do not appear in the keys or the container operations:

sage: f = Family([3,4,7], lambda i: 2*i, hidden_keys=[2])
sage: f
Finite family {3: 6, 4: 8, 7: 14}
sage: f.keys()
[3, 4, 7]
sage: f.hidden_keys()
[2]
sage: f[7]
14
sage: f[2]
4
sage: list(f)
[6, 8, 14]
sage: [x for x in f]
[6, 8, 14]
sage: len(f)
3

>>> from sage.all import *
>>> f = Family([Integer(3),Integer(4),Integer(7)], lambda i: Integer(2)*i, hidden_keys=[Integer(2)])
>>> f
Finite family {3: 6, 4: 8, 7: 14}
>>> f.keys()
[3, 4, 7]
>>> f.hidden_keys()
[2]
>>> f[Integer(7)]
14
>>> f[Integer(2)]
4
>>> list(f)
[6, 8, 14]
>>> [x for x in f]
[6, 8, 14]
>>> len(f)
3

The following example illustrates when the function is actually called:

sage: def compute_value(i):
....:     print('computing 2*'+str(i))
....:     return 2*i
sage: f = Family([3,4,7], compute_value, hidden_keys=[2])
computing 2*3
computing 2*4
computing 2*7
sage: f
Finite family {3: 6, 4: 8, 7: 14}
sage: f.keys()
[3, 4, 7]
sage: f.hidden_keys()
[2]
sage: f[7]
14
sage: f[2]
computing 2*2
4
sage: f[2]
4
sage: list(f)
[6, 8, 14]
sage: [x for x in f]
[6, 8, 14]
sage: len(f)
3

>>> from sage.all import *
>>> def compute_value(i):
...     print('computing 2*'+str(i))
...     return Integer(2)*i
>>> f = Family([Integer(3),Integer(4),Integer(7)], compute_value, hidden_keys=[Integer(2)])
computing 2*3
computing 2*4
computing 2*7
>>> f
Finite family {3: 6, 4: 8, 7: 14}
>>> f.keys()
[3, 4, 7]
>>> f.hidden_keys()
[2]
>>> f[Integer(7)]
14
>>> f[Integer(2)]
computing 2*2
4
>>> f[Integer(2)]
4
>>> list(f)
[6, 8, 14]
>>> [x for x in f]
[6, 8, 14]
>>> len(f)
3

Here is a close variant where the function for the hidden keys is different from that for the other keys:

sage: f = Family([3,4,7], lambda i: 2*i, hidden_keys=[2], hidden_function = lambda i: 3*i)
sage: f
Finite family {3: 6, 4: 8, 7: 14}
sage: f.keys()
[3, 4, 7]
sage: f.hidden_keys()
[2]
sage: f[7]
14
sage: f[2]
6
sage: list(f)
[6, 8, 14]
sage: [x for x in f]
[6, 8, 14]
sage: len(f)
3

>>> from sage.all import *
>>> f = Family([Integer(3),Integer(4),Integer(7)], lambda i: Integer(2)*i, hidden_keys=[Integer(2)], hidden_function = lambda i: Integer(3)*i)
>>> f
Finite family {3: 6, 4: 8, 7: 14}
>>> f.keys()
[3, 4, 7]
>>> f.hidden_keys()
[2]
>>> f[Integer(7)]
14
>>> f[Integer(2)]
6
>>> list(f)
[6, 8, 14]
>>> [x for x in f]
[6, 8, 14]
>>> len(f)
3

Family accept finite and infinite EnumeratedSets as input:

sage: f = Family(FiniteEnumeratedSet([1,2,3]))
sage: f
Family (1, 2, 3)
sage: f = Family(NonNegativeIntegers())
sage: f
Family (Non negative integers)

>>> from sage.all import *
>>> f = Family(FiniteEnumeratedSet([Integer(1),Integer(2),Integer(3)]))
>>> f
Family (1, 2, 3)
>>> f = Family(NonNegativeIntegers())
>>> f
Family (Non negative integers)

sage: f = Family(FiniteEnumeratedSet([3,4,7]), lambda i: 2*i)
sage: f
Finite family {3: 6, 4: 8, 7: 14}
sage: f.keys()
{3, 4, 7}
sage: f[7]
14
sage: list(f)
[6, 8, 14]
sage: [x for x in f]
[6, 8, 14]
sage: len(f)
3

>>> from sage.all import *
>>> f = Family(FiniteEnumeratedSet([Integer(3),Integer(4),Integer(7)]), lambda i: Integer(2)*i)
>>> f
Finite family {3: 6, 4: 8, 7: 14}
>>> f.keys()
{3, 4, 7}
>>> f[Integer(7)]
14
>>> list(f)
[6, 8, 14]
>>> [x for x in f]
[6, 8, 14]
>>> len(f)
3
class sage.sets.family.FiniteFamily[source]#

Bases: AbstractFamily

A FiniteFamily is an associative container which models a finite family \((f_i)_{i \in I}\). Its elements \(f_i\) are therefore its values. Instances should be created via the Family() factory. See its documentation for examples and tests.

EXAMPLES:

We define the family \((f_i)_{i \in \{3,4,7\}}\) with \(f_3=a\), \(f_4=b\), and \(f_7=d\):

sage: from sage.sets.family import FiniteFamily
sage: f = FiniteFamily({3: 'a', 4: 'b', 7: 'd'})

>>> from sage.all import *
>>> from sage.sets.family import FiniteFamily
>>> f = FiniteFamily({Integer(3): 'a', Integer(4): 'b', Integer(7): 'd'})

Individual elements are accessible as in a usual dictionary:

sage: f[7]
'd'

>>> from sage.all import *
>>> f[Integer(7)]
'd'

And the other usual dictionary operations are also available:

sage: len(f)
3
sage: f.keys()
[3, 4, 7]

>>> from sage.all import *
>>> len(f)
3
>>> f.keys()
[3, 4, 7]

However f behaves as a container for the \(f_i\)’s:

sage: list(f)
['a', 'b', 'd']
sage: [ x for x in f ]
['a', 'b', 'd']

>>> from sage.all import *
>>> list(f)
['a', 'b', 'd']
>>> [ x for x in f ]
['a', 'b', 'd']

The order of the elements can be specified using the keys optional argument:

sage: f = FiniteFamily({"a": "aa", "b": "bb", "c" : "cc" }, keys = ["c", "a", "b"])
sage: list(f)
['cc', 'aa', 'bb']

>>> from sage.all import *
>>> f = FiniteFamily({"a": "aa", "b": "bb", "c" : "cc" }, keys = ["c", "a", "b"])
>>> list(f)
['cc', 'aa', 'bb']
cardinality()[source]#

Returns the number of elements in self.

EXAMPLES:

sage: from sage.sets.family import FiniteFamily
sage: f = FiniteFamily({3: 'a', 4: 'b', 7: 'd'})
sage: f.cardinality()
3

>>> from sage.all import *
>>> from sage.sets.family import FiniteFamily
>>> f = FiniteFamily({Integer(3): 'a', Integer(4): 'b', Integer(7): 'd'})
>>> f.cardinality()
3
has_key(k)[source]#

Returns whether k is a key of self

EXAMPLES:

sage: Family({"a":1, "b":2, "c":3}).has_key("a")
True
sage: Family({"a":1, "b":2, "c":3}).has_key("d")
False

>>> from sage.all import *
>>> Family({"a":Integer(1), "b":Integer(2), "c":Integer(3)}).has_key("a")
True
>>> Family({"a":Integer(1), "b":Integer(2), "c":Integer(3)}).has_key("d")
False
keys()[source]#

Returns the index set of this family

EXAMPLES:

sage: f = Family(["c", "a", "b"], lambda x: x+x)
sage: f.keys()
['c', 'a', 'b']

>>> from sage.all import *
>>> f = Family(["c", "a", "b"], lambda x: x+x)
>>> f.keys()
['c', 'a', 'b']
values()[source]#

Returns the elements of this family

EXAMPLES:

sage: f = Family(["c", "a", "b"], lambda x: x+x)
sage: f.values()
['cc', 'aa', 'bb']

>>> from sage.all import *
>>> f = Family(["c", "a", "b"], lambda x: x+x)
>>> f.values()
['cc', 'aa', 'bb']
class sage.sets.family.FiniteFamilyWithHiddenKeys(dictionary, hidden_keys, hidden_function, keys=None)[source]#

Bases: FiniteFamily

A close variant of FiniteFamily where the family contains some hidden keys whose corresponding values are computed lazily (and remembered). Instances should be created via the Family() factory. See its documentation for examples and tests.

Caveat: Only instances of this class whose functions are compatible with sage.misc.fpickle can be pickled.

hidden_keys()[source]#

Returns self’s hidden keys.

EXAMPLES:

sage: f = Family([3,4,7], lambda i: 2*i, hidden_keys=[2])
sage: f.hidden_keys()
[2]

>>> from sage.all import *
>>> f = Family([Integer(3),Integer(4),Integer(7)], lambda i: Integer(2)*i, hidden_keys=[Integer(2)])
>>> f.hidden_keys()
[2]
class sage.sets.family.LazyFamily(set, function, name=None)[source]#

Bases: AbstractFamily

A LazyFamily(I, f) is an associative container which models the (possibly infinite) family \((f(i))_{i \in I}\).

Instances should be created via the Family() factory. See its documentation for examples and tests.

cardinality()[source]#

Return the number of elements in self.

EXAMPLES:

sage: from sage.sets.family import LazyFamily
sage: f = LazyFamily([3,4,7], lambda i: 2*i)
sage: f.cardinality()
3
sage: l = LazyFamily(NonNegativeIntegers(), lambda i: 2*i)
sage: l.cardinality()
+Infinity

>>> from sage.all import *
>>> from sage.sets.family import LazyFamily
>>> f = LazyFamily([Integer(3),Integer(4),Integer(7)], lambda i: Integer(2)*i)
>>> f.cardinality()
3
>>> l = LazyFamily(NonNegativeIntegers(), lambda i: Integer(2)*i)
>>> l.cardinality()
+Infinity
keys()[source]#

Returns self’s keys.

EXAMPLES:

sage: from sage.sets.family import LazyFamily
sage: f = LazyFamily([3,4,7], lambda i: 2*i)
sage: f.keys()
[3, 4, 7]

>>> from sage.all import *
>>> from sage.sets.family import LazyFamily
>>> f = LazyFamily([Integer(3),Integer(4),Integer(7)], lambda i: Integer(2)*i)
>>> f.keys()
[3, 4, 7]
class sage.sets.family.TrivialFamily(enumeration)[source]#

Bases: AbstractFamily

TrivialFamily turns a list/tuple \(c\) into a family indexed by the set \(\{0, \dots, |c|-1\}\).

Instances should be created via the Family() factory. See its documentation for examples and tests.

cardinality()[source]#

Return the number of elements in self.

EXAMPLES:

sage: from sage.sets.family import TrivialFamily
sage: f = TrivialFamily([3,4,7])
sage: f.cardinality()
3

>>> from sage.all import *
>>> from sage.sets.family import TrivialFamily
>>> f = TrivialFamily([Integer(3),Integer(4),Integer(7)])
>>> f.cardinality()
3
keys()[source]#

Returns self’s keys.

EXAMPLES:

sage: from sage.sets.family import TrivialFamily
sage: f = TrivialFamily([3,4,7])
sage: f.keys()
[0, 1, 2]

>>> from sage.all import *
>>> from sage.sets.family import TrivialFamily
>>> f = TrivialFamily([Integer(3),Integer(4),Integer(7)])
>>> f.keys()
[0, 1, 2]
map(f, name=None)[source]#

Return the family \(( f(\mathtt{self}[i]) )_{i \in I}\), where \(I\) is the index set of self.

The result is again a TrivialFamily.

EXAMPLES:

sage: from sage.sets.family import TrivialFamily
sage: f = TrivialFamily(['a', 'b', 'd'])
sage: g = f.map(lambda x: x + '1'); g
Family ('a1', 'b1', 'd1')

>>> from sage.all import *
>>> from sage.sets.family import TrivialFamily
>>> f = TrivialFamily(['a', 'b', 'd'])
>>> g = f.map(lambda x: x + '1'); g
Family ('a1', 'b1', 'd1')